def getgroup(m1,ell):
pind = {2: 0,3:1,5:2,7:3,11:4,13:5}
if len(m1[3][2])==0:
return [m1[2], m1[0]]
myA = list2string(m1[3][0])
myB = list2string(m1[3][1])
if len(myA)==0 and len(myB)==0:
return [small_group_display_knowl(1,1,db()), 1]
newthing = familydb().find_one({'A': myA, 'B': myB})
if newthing is None:
return ['??', 1]
newthing = newthing['mono'][pind[ell]]
newthing[1] = dogapthing(newthing[1])
return [newthing[1][2], newthing[1][0]]
def getgroup(m1,ell):
pind = {2: 0,3:1,5:2,7:3,11:4,13:5}
if len(m1[3][2])==0:
return [m1[2], m1[0]]
myA = list2string(m1[3][0])
myB = list2string(m1[3][1])
if len(myA)==0 and len(myB)==0:
return [small_group_display_knowl(1,1), 1]
mono = db.hgm_families.lucky({'A': myA, 'B': myB}, projection="mono")
if mono is None:
return ['??', 1]
newthing = mono[pind[ell]]
newthing = dogapthing(newthing[1])
return [newthing[2], newthing[0]]
def dogapthing(m1):
mnew = str(m1[2])
mnew = mnew.replace(' ','')
if GAP_ID_RE.match(mnew):
mnew = mnew[1:-1]
two = mnew.split(',')
two = [int(j) for j in two]
try:
m1[2] = small_group_display_knowl(two[0],two[1],db())
except TypeError:
m1[2] = 'Gap[%d,%d]' % (two[0],two[1])
else:
m1[2] = '$%s$'% m1[2]
return m1
def dogapthing(m1):
mnew = str(m1[2])
mnew = mnew.replace(' ','')
if GAP_ID_RE.match(mnew):
mnew = mnew[1:-1]
two = mnew.split(',')
two = [int(j) for j in two]
try:
m1[2] = small_group_display_knowl(two[0],two[1])
except TypeError:
m1[2] = 'Gap[%d,%d]' % (two[0],two[1])
else:
m1[2] = '$%s$'% m1[2]
return m1
def getgroup(m1,ell):
pind = {2: 0,3:1,5:2,7:3,11:4,13:5}
if len(m1[3][2])==0:
return [m1[2], m1[0]]
myA = list2string(m1[3][0])
myB = list2string(m1[3][1])
if len(myA)==0 and len(myB)==0:
return [small_group_display_knowl(1,1,db()), 1]
newthing = familydb().find_one({'A': myA, 'B': myB})
if newthing is None:
return ['??', 1]
newthing = newthing['mono'][pind[ell]]
newthing[1] = dogapthing(newthing[1])
return [newthing[1][2], newthing[1][0]]
def dogapthing(m1):
mnew = str(m1[2])
mnew = mnew.replace(' ', '')
if GAP_ID_RE.match(mnew):
mnew = mnew[1:-1]
two = mnew.split(',')
two = [int(j) for j in two]
try:
m1[2] = small_group_display_knowl(two[0], two[1])
except TypeError:
m1[2] = 'Gap[%d,%d]' % (two[0], two[1])
else:
# Fix multiple backslashes
m1[2] = re.sub(r'\\+', r'\\', m1[2])
m1[2] = '$%s$' % m1[2]
return m1
def render_group_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
label = label.replace('t', 'T')
C = base.getDBConnection()
data = C.transitivegroups.groups.find_one({'label': label})
if data is None:
bread = get_bread([("Search Error", ' ')])
info['err'] = "Group " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
data['label_raw'] = label.lower()
title = 'Galois Group: ' + label
wgg = WebGaloisGroup.from_data(data)
n = data['n']
t = data['t']
data['yesno'] = yesno
order = data['order']
data['orderfac'] = latex(ZZ(order).factor())
orderfac = latex(ZZ(order).factor())
data['ordermsg'] = "$%s=%s$" % (order, latex(orderfac))
if order == 1:
data['ordermsg'] = "$1$"
if ZZ(order).is_prime():
data['ordermsg'] = "$%s$ (is prime)" % order
pgroup = len(ZZ(order).prime_factors()) < 2
if n == 1:
G = gap.SmallGroup(n, t)
else:
G = gap.TransitiveGroup(n, t)
if ZZ(order) < ZZ('10000000000'):
ctable = chartable(n, t)
else:
ctable = 'Group too large'
data['gens'] = generators(n, t)
if n == 1 and t == 1:
data['gens'] = 'None needed'
data['chartable'] = ctable
data['parity'] = "$%s$" % data['parity']
data['cclasses'] = conjclasses(G, n)
data['subinfo'] = subfield_display(C, n, data['subs'])
data['resolve'] = resolve_display(C, data['resolve'])
if data['gapid'] == 0:
data['gapid'] = "Data not available"
else:
data['gapid'] = small_group_display_knowl(int(data['order']),int(data['gapid']),C, str([int(data['order']),int(data['gapid'])]))
data['otherreps'] = wgg.otherrep_list()
ae = wgg.arith_equivalent()
if ae>0:
if ae>1:
data['arith_equiv'] = r'A number field with this Galois group has %d <a knowl="nf.arithmetically_equivalent", title="arithmetically equivalent">arithmetically equivalent</a> fields.'% ae
else:
data['arith_equiv'] = r'A number field with this Galois group has exactly one <a knowl="nf.arithmetically_equivalent", title="arithmetically equivalent">arithmetically equivalent</a> field.'
else:
data['arith_equiv'] = r'A number field with this Galois group has no <a knowl="nf.arithmetically_equivalent", title="arithmetically equivalent">arithmetically equivalent</a> fields.'
if len(data['otherreps']) == 0:
data['otherreps']="There is no other low degree representation."
query={'galois': bson.SON([('n', n), ('t', t)])}
C = base.getDBConnection()
intreps = C.transitivegroups.Gmodules.find({'n': n, 't': t}).sort('index', pymongo.ASCENDING)
# turn cursor into a list
intreps = [z for z in intreps]
if len(intreps) > 0:
data['int_rep_classes'] = [str(z[0]) for z in intreps[0]['gens']]
for onerep in intreps:
onerep['gens']=[list_to_latex_matrix(z[1]) for z in onerep['gens']]
data['int_reps'] = intreps
data['int_reps_complete'] = int_reps_are_complete(intreps)
dcq = data['moddecompuniq']
if dcq[0] == 0:
data['decompunique'] = 0
else:
data['decompunique'] = dcq[0]
data['isoms'] = [[mult2mult(z[0]), mult2mult(z[1])] for z in dcq[1]]
data['isoms'] = [[modules2string(n,t,z[0]), modules2string(n,t,z[1])] for z in data['isoms']]
#print dcq[1]
#print data['isoms']
friends = []
one = C.numberfields.fields.find_one(query)
if one:
friends.append(('Number fields with this Galois group', url_for('number_fields.number_field_render_webpage')+"?galois_group=%dT%d" % (n, t) ))
prop2 = [('Label', label),
('Order', '\(%s\)' % order),
('n', '\(%s\)' % data['n']),
('Cyclic', yesno(data['cyc'])),
('Abelian', yesno(data['ab'])),
('Solvable', yesno(data['solv'])),
('Primitive', yesno(data['prim'])),
('$p$-group', yesno(pgroup)),
]
pretty = group_display_pretty(n,t,C)
if len(pretty)>0:
prop2.extend([('Group:', pretty)])
info['pretty_name'] = pretty
data['name'] = re.sub(r'_(\d+)',r'_{\1}',data['name'])
data['name'] = re.sub(r'\^(\d+)',r'^{\1}',data['name'])
info.update(data)
bread = get_bread([(label, ' ')])
return render_template("gg-show-group.html", credit=GG_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends)
def render_group_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
label = label.replace('t', 'T')
C = base.getDBConnection()
data = C.transitivegroups.groups.find_one({'label': label})
if data is None:
bread = get_bread([("Search Error", ' ')])
info['err'] = "Group " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
data['label_raw'] = label.lower()
title = 'Galois Group: ' + label
wgg = WebGaloisGroup.from_data(data)
n = data['n']
t = data['t']
data['yesno'] = yesno
order = data['order']
data['orderfac'] = latex(ZZ(order).factor())
orderfac = latex(ZZ(order).factor())
data['ordermsg'] = "$%s=%s$" % (order, latex(orderfac))
if order == 1:
data['ordermsg'] = "$1$"
if ZZ(order).is_prime():
data['ordermsg'] = "$%s$ (is prime)" % order
pgroup = len(ZZ(order).prime_factors()) < 2
if n == 1:
G = gap.SmallGroup(n, t)
else:
G = gap.TransitiveGroup(n, t)
if ZZ(order) < ZZ('10000000000'):
ctable = chartable(n, t)
else:
ctable = 'Group too large'
data['gens'] = generators(n, t)
if n == 1 and t == 1:
data['gens'] = 'None needed'
data['chartable'] = ctable
data['parity'] = "$%s$" % data['parity']
data['cclasses'] = conjclasses(G, n)
data['subinfo'] = subfield_display(C, n, data['subs'])
data['resolve'] = resolve_display(C, data['resolve'])
if data['gapid'] == 0:
data['gapid'] = "Data not available"
else:
data['gapid'] = small_group_display_knowl(int(data['order']),int(data['gapid']),C, str([int(data['order']),int(data['gapid'])]))
data['otherreps'] = wgg.otherrep_list()
ae = wgg.arith_equivalent()
if ae>0:
if ae>1:
data['arith_equiv'] = r'A number field with this Galois group has %d <a knowl="nf.arithmetically_equivalent", title="arithmetically equivalent">arithmetically equivalent</a> fields.'% ae
else:
data['arith_equiv'] = r'A number field with this Galois group has exactly one <a knowl="nf.arithmetically_equivalent", title="arithmetically equivalent">arithmetically equivalent</a> field.'
else:
data['arith_equiv'] = r'A number field with this Galois group has no <a knowl="nf.arithmetically_equivalent", title="arithmetically equivalent">arithmetically equivalent</a> fields.'
if len(data['otherreps']) == 0:
data['otherreps']="There is no other low degree representation."
query={'galois': bson.SON([('n', n), ('t', t)])}
C = base.getDBConnection()
intreps = C.transitivegroups.Gmodules.find({'n': n, 't': t}).sort('index', pymongo.ASCENDING)
# turn cursor into a list
intreps = [z for z in intreps]
if len(intreps) > 0:
data['int_rep_classes'] = [str(z[0]) for z in intreps[0]['gens']]
for onerep in intreps:
onerep['gens']=[list_to_latex_matrix(z[1]) for z in onerep['gens']]
data['int_reps'] = intreps
data['int_reps_complete'] = int_reps_are_complete(intreps)
dcq = data['moddecompuniq']
if dcq[0] == 0:
data['decompunique'] = 0
else:
data['decompunique'] = dcq[0]
data['isoms'] = [[mult2mult(z[0]), mult2mult(z[1])] for z in dcq[1]]
data['isoms'] = [[modules2string(n,t,z[0]), modules2string(n,t,z[1])] for z in data['isoms']]
#print dcq[1]
#print data['isoms']
friends = []
one = C.numberfields.fields.find_one(query)
if one:
friends.append(('Number fields with this Galois group', url_for('number_fields.number_field_render_webpage')+"?galois_group=%dT%d" % (n, t) ))
prop2 = [('Label', label),
('Order', '\(%s\)' % order),
('n', '\(%s\)' % data['n']),
('Cyclic', yesno(data['cyc'])),
('Abelian', yesno(data['ab'])),
('Solvable', yesno(data['solv'])),
('Primitive', yesno(data['prim'])),
('$p$-group', yesno(pgroup)),
]
pretty = group_display_pretty(n,t,C)
if len(pretty)>0:
prop2.extend([('Group:', pretty)])
info['pretty_name'] = pretty
data['name'] = re.sub(r'_(\d+)',r'_{\1}',data['name'])
data['name'] = re.sub(r'\^(\d+)',r'^{\1}',data['name'])
info.update(data)
bread = get_bread([(label, ' ')])
return render_template("gg-show-group.html", credit=GG_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends)
def render_hgm_family_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
data = familydb().find_one({'label': label})
if data is None:
bread = get_bread([("Search error", url_for('.search'))])
info['err'] = "Family of hypergeometric motives " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Hypergeometric Motive Family:' + label
A = string2list(data['A'])
B = string2list(data['B'])
hodge = data['famhodge']
mydet = data['det']
detexp = QQ(data['weight']*data['degree'])
detexp = -detexp/2
mydet = r'\Q(%s)\otimes\Q(\sqrt{'%str(detexp)
if int(data['det'][0]) != 1:
mydet += str(data['det'][0])
if len(data['det'][1])>0:
mydet += data['det'][1]
if int(data['det'][0]) == 1 and len(data['det'][1])==0:
mydet += '1'
mydet += '})'
bezoutmat = matrix(data['bezout'])
bezoutdet = bezoutmat.det()
bezoutmat = latex(bezoutmat)
snf = string2list(data['snf'])
snf = list2Cnstring(snf)
typee = 'Orthogonal'
if (data['weight'] % 2) == 1 and (data['degree'] % 2) == 0:
typee = 'Symplectic'
ppart = [[2, [string2list(u) for u in [data['A2'],data['B2'],data['C2']]]],
[3, [string2list(u) for u in [data['A3'],data['B3'],data['C3']]]],
[5, [string2list(u) for u in [data['A5'],data['B5'],data['C5']]]],
[7, [string2list(u) for u in [data['A7'],data['B7'],data['C7']]]]]
prop2 = [
('Degree', '\(%s\)' % data['degree']),
('Weight', '\(%s\)' % data['weight'])
]
mono = [[m[0], dogapthing(m[1]),
getgroup(m[1],m[0]),
latex(ZZ(m[1][0]).factor())] for m in data['mono']]
mono = [[m[0], m[1], m[2][0], splitint(m[1][0]/m[2][1],m[0]), m[3]] for m in mono]
info.update({
'A': A,
'B': B,
'degree': data['degree'],
'weight': data['weight'],
'hodge': hodge,
'det': mydet,
'snf': snf,
'bezoutmat': bezoutmat,
'bezoutdet': bezoutdet,
'mono': mono,
'imprim': data['imprim'],
'ppart': ppart,
'type': typee,
'junk': small_group_display_knowl(18,2,db()),
'showlist': showlist
})
friends = [('Motives in the family', url_for('hypergm.index')+"?A=%s&B=%s" % (str(A), str(B)))]
# if unramfriend != '':
# friends.append(('Unramified subfield', unramfriend))
# if rffriend != '':
# friends.append(('Discriminant root field', rffriend))
info.update({"plotcircle": url_for(".hgm_family_circle_image", AB = "A"+".".join(map(str,A))+"_B"+".".join(map(str,B)))})
info.update({"plotlinear": url_for(".hgm_family_linear_image", AB = "A"+".".join(map(str,A))+"_B"+".".join(map(str,B)))})
info.update({"plotconstant": url_for(".hgm_family_constant_image", AB = "A"+".".join(map(str,A))+"_B"+".".join(map(str,B)))})
bread = get_bread([(label, ' ')])
return render_template("hgm-show-family.html", credit=HGM_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends, learnmore=learnmore_list())
def render_hgm_family_webpage(label):
data = None
info = {}
data = db.hgm_families.lookup(label)
if data is None:
abort(404, "Hypergeometric motive family " + label + " was not found in the database.")
title = 'Hypergeometric Motive Family:' + label
A = data['A']
B = data['B']
hodge = data['famhodge']
mydet = data['det']
detexp = QQ(data['weight']*data['degree'])
detexp = -detexp/2
mydet = r'\Q(%s)\otimes\Q(\sqrt{'%str(detexp)
if int(data['det'][0]) != 1:
mydet += str(data['det'][0])
if len(data['det'][1])>0:
mydet += data['det'][1]
if int(data['det'][0]) == 1 and len(data['det'][1])==0:
mydet += '1'
mydet += '})'
bezoutmat = matrix(data['bezout'])
bezoutdet = bezoutmat.det()
bezoutmat = latex(bezoutmat)
snf = data['snf']
snf = list2Cnstring(snf)
typee = 'Orthogonal'
if (data['weight'] % 2) == 1 and (data['degree'] % 2) == 0:
typee = 'Symplectic'
ppart = [[2, [data['A2'],data['B2'],data['C2']]],
[3, [data['A3'],data['B3'],data['C3']]],
[5, [data['A5'],data['B5'],data['C5']]],
[7, [data['A7'],data['B7'],data['C7']]]]
prop2 = [
('Degree', '\(%s\)' % data['degree']),
('Weight', '\(%s\)' % data['weight'])
]
mono = [m for m in data['mono'] if m[1] != 0]
mono = [[m[0], dogapthing(m[1]),
getgroup(m[1],m[0]),
latex(ZZ(m[1][0]).factor())] for m in mono]
mono = [[m[0], m[1], m[2][0], splitint(m[1][0]/m[2][1],m[0]), m[3]] for m in mono]
info.update({
'A': A,
'B': B,
'degree': data['degree'],
'weight': data['weight'],
'hodge': hodge,
'det': mydet,
'snf': snf,
'bezoutmat': bezoutmat,
'bezoutdet': bezoutdet,
'mono': mono,
'imprim': data['imprim'],
'ppart': ppart,
'type': typee,
'junk': small_group_display_knowl(18,2),
'showlist': showlist
})
friends = [('Motives in the family', url_for('hypergm.index')+"?A=%s&B=%s" % (str(A), str(B)))]
# if unramfriend != '':
# friends.append(('Unramified subfield', unramfriend))
# if rffriend != '':
# friends.append(('Discriminant root field', rffriend))
info.update({"plotcircle": url_for(".hgm_family_circle_image", AB = "A"+".".join(map(str,A))+"_B"+".".join(map(str,B)))})
info.update({"plotlinear": url_for(".hgm_family_linear_image", AB = "A"+".".join(map(str,A))+"_B"+".".join(map(str,B)))})
info.update({"plotconstant": url_for(".hgm_family_constant_image", AB = "A"+".".join(map(str,A))+"_B"+".".join(map(str,B)))})
bread = get_bread([(label, ' ')])
return render_template("hgm-show-family.html", credit=HGM_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends, learnmore=learnmore_list())